Question

The back wall of an auditorium is 26.0 m from the stage. If you are seated in the middle row, how much time elapses between a sound from the stage reaching your ear directly and the same sound reaching your ear after reflecting from the back wall?

Answer 1

The time elapsed is
`0.08 s`

Step-by-step explanation:

In order to get a good we need to understand that the concept used in this problem is the relation of time, speed and distance.

Initially, calculate the time taken by direct sound and the reflected sound to reach at the middle row using the relation of time, speed and distance. Finally, calculate the elapsed time by subtracting the time taken by direct sound from the time taken by reflected sound. let look at some fundamentals

Time, distance and speed:

The relation between the speed, time and the distance is,

`t = (d)/(v)`

Here,
`t` is the time, is the distance and
`v` is the speed.

Step 1 of 2

Distance traveled by direct sound from the wall to the middle row is
`(L)/(2)`

Here,
`L` is the distance of the wall from the stage.

The time taken by direct sound from the wall to the middle row is,

`t_(d) = (d_d)/(v)`

Substitute
`(L)/(2)` for
`d_d`

`t_(d) = ((L)/(2) )/(v)`

`(L)/(2v) -----(1)`

Distance traveled by reflected sound from the wall to the middle row is
`[L +(L)/(2) ]`

The time taken by reflected sound from the wall to the middle row is,

`t_(r) = (d_r)/(v)`

Substitute
`[L +(L)/(2) ]` for
`d_r`

`t_(r) = ([L+(L)/(2) ])/(v)`

`= [((3L)/(2) )/(v) ]`

`t_(r) = (3L)/(2v) ----(2)`

Note: The time taken by direct sound is calculated using the distance traveled by direct sound from the wall to the middle row. The distance traveled by the reflected sound is the sum of distance traveled by the direct sound and the distance between the wall and the stage.

Step 2 of 2

The elapsed time calculated as,

`t = t_r +t_d -----(3)`

Substitute equation (1) and equation (2) in equation (3).

`t = (3L)/(v)+ (L)/(2v)`

`= (L)/(v)`

Substitute 343 m/s for
`v` and 26 m for
`L` in the above expression.

`t = (26m)/(343m/s)`

`=0.08 s`

Note

The elapsed time is calculated by subtracting the time taken by direct sound from the time taken by reflected sound. Then, the value corresponding to the distance between wall and the stage and the value corresponding to the speed of sound are substituted in the expression of time.

Answer 2

Δt=0.076s

Step-by-step explanation:

Assuming that the speed of sound is 343 m/s at 20°C the time elapsed from the stage to you in the middle row is the distance divided by the speed of sound as:

`(13m)/(343m/s)=0.0379s\\`

The time elapsed for the same sound to get the reflected from back wall and reach your ear expressed as

`(26m+13m)/(343m/s)=0.1137s`

Hence we have difference in two times as:

`0.1137s-0.0379s=0.076s`

Δt=0.076s